Linguist and Philos (2007) 30:669–690
DOI 10.1007/s10988-008-9029-y
RESEARCH ARTICLE
Structurally-defined alternatives
Roni Katzir
Published online: 5 July 2008
Ó Springer Science+Business Media B.V. 2008
Abstract Scalar implicatures depend on alternatives in order to avoid the
symmetry problem. I argue for a structure-sensitive characterization of these
alternatives: the alternatives for a structure are all those structures that are at most as
complex as the original one. There have been claims in the literature that complexity is irrelevant for implicatures and that the relevant condition is the semantic
notion of monotonicity. I provide new data that pose a challenge to the use of
monotonicity and that support the structure-sensitive definition. I show that what
appeared to be a problem for the complexity approach is overcome once an
appropriate notion of complexity is adopted, and that upon closer inspection, the
argument in favor of monotonicity turns out to be an argument against it and in
favor of the complexity approach.
Keywords Scalar implicature Horn scales Complexity Alternatives
Computing the scalar implicatures of a linguistic object involves reference to
alternative objects that were not used. This paper argues for a structure-sensitive
characterization of these alternative objects. For any structure /, the alternatives
will be all those structures that are at most as complex as /, under a particular
notion of complexity:
(1)
Astr ð/Þ ¼ f/0 j /0 . /g
R. Katzir (&)
Linguistics and Philosophy, MIT, 77 Mass. Ave., Bldg. 32-D808,
Cambridge, MA 02139, USA
e-mail: [email protected]
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Complexity is not a new notion in the domain of conversational reasoning. It is
present in Grice’s Maxim of Manner and elsewhere. However, there has also been
skepticism about the relevance of complexity for scalar implicatures.1 One reason
for skepticism is the scarcity of instances where an inference about / seems to
require reasoning about a strictly simpler /0 . In most cases, / and /0 are of roughly
the same complexity. This is surprising under (1). On the other hand, under the
widely assumed characterization of alternatives in terms of scales (Horn 1972;
Gazdar 1979), the scarcity of complexity-related implicatures makes much more
sense. And there have been other objections. Perhaps the most serious objection to
the use of complexity is an argument by Matsumoto (1995), who observes that in
certain cases an inference about an utterance / requires reference to an alternative
that appears to be strictly more complex than /. Matsumoto’s conclusion is that
complexity does not play a role in the computation of implicatures. Instead, he
follows Horn (1989) in adopting a semantic constraint on scale-mates that makes
use of the notion of monotonicity. We will review these arguments in Sect. 1.
Section 2 presents an empirical observation that will suggest that complexity
might be the right way to go after all. The observation is this: in certain contexts, a
complex, non-monotonic expression can give rise to an implicature that requires a
comparison with a simpler alternative. This goes against the monotonicity condition, and indirectly also against the notion of scale, while supporting the complexity
approach.
To turn the complexity approach into an actual proposal, we will need an explicit
definition of structural complexity. We will also have to address Matsumoto’s
objection, as well as the concern about the scarcity of comparisons with strictly
simpler alternatives. Section 3 offers a definition of complexity, based on the idea
that we can transform / into a structure that is no more complex if we restrict
ourselves to (a) deleting elements in /, and (b) substituting elements in / with
other elements from an appropriately defined source. In Sect. 4 we will see that a
large family of conversational inferences is correctly predicted, including certain
disjunction puzzles analyzed by Sauerland (2004), as well as some new cases.
Along the way we will see why inferences that refer to simpler alternatives are rare:
usually, simpler structures result in weaker assertions, so no inference arises. Once
this is observed, the problem can be avoided by embedding complex structures in
downward-entailing contexts, where entailment relations are reversed. As predicted,
we will find that in such contexts the relevant inferences arise. In Sect. 5 we turn to
Matsumoto’s argument against complexity. Addressing the argument will force us
to be more careful about our notion of substitution source. This, in turn, will lead to
certain new predictions.
The primary goal of this paper is to argue for a structure-sensitive characterization of the alternatives. The secondary goals are to argue against a monotonicity
constraint on alternatives, as well as against the intermediate step of scales. Before
we start, I would like to mention what this paper is not about. I will have very little
to say about the question of whether implicatures are computed locally (as in
1
From here on I will simply write implicatures rather than scalar implicatures, hoping that no
confusion will arise.
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Landman 2000; Chierchia 2004; Fox 2006, and others) or globally (as in the
traditional accounts, as well as in recent proposals by Sauerland 2004; Russell
2006). Much of the discussion will be framed in terms of global, neo-Gricean
reasoning, but this is done for ease of presentation only. As far as I can tell very
little of the current discussion depends on this choice. Secondly, I will avoid talking
about the relation of implicatures to exhaustivity in answers, as in the proposals of
van Rooij and Schulz (2004), Sevi (2005), Spector (2007), and others. Finally, I will
have nothing to say about implicatures that depend on context and world knowledge, of the kind discussed by Hirschberg (1985/1991).
1 Background
1.1 Conversational reasoning
Informally speaking, a cooperative speaker can be assumed to make maximally
helpful contributions to the conversation. If / and /0 are potential contributions, and
if /0 is strictly better than / (i.e., /0 is at least as good as / in every relevant respect,
and strictly better than it in at least one relevant respect), then a cooperative speaker
will prefer using /0 when possible. Consider, for example, a situation in which (2)
and (3) are both potential contributions to the conversation.
(2)
John ate some of the cake
(3)
John ate all of the cake
(3) is more informative than (2), but in all other respects the two sentences seem
equally good. A cooperative speaker, then, will prefer using (3) to (2). If despite this
preference the speaker has uttered (2), the hearer can conclude that there were
considerations other than those mentioned above that prevented the speaker from
using (3). Depending on what else can be assumed, the hearer may take this
inference to license further conclusions. For example, they can conclude that the
speaker does not have an opinion about (3), or that the speaker believes (3) to be
false. Still in loose terms, the preference can be stated like this:
(4)
Conversational principle (informal version): do not use a linguistic object /
if there is another object /0 such that both
a. /0 is better than /, and
b. /0 could have been used
Several aspects of (4) must be specified if we want to be able to apply it at all. First,
one has to decide what kinds of objects / and its alternatives are. Then one has to
provide concrete content to the term use (in fact, we will soon see that it is common
to provide different content to each of the two occurrences of use), as well as to the
term better. Once this is done, (4) can license the following kind of inference: if a
cooperative speaker has used /, then for all better alternatives /0 it is not the case
that the speaker could have used /0 instead of /. In our example, since (3) is better
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than (2), the use of (2) licenses the inference that it is not the case that the speaker
could have used (3).
Depending on what else is assumed, these inferences can sometimes license
further inferences about why it is that the better /0 could not be used. For example,
if the speaker can be assumed to be in a position to use either /0 or :/0 , the weak
inference it is not the case that the speaker could have used /0 can be strengthened
into it is the case that the speaker could have used :/0 . If such an assumption can
be made about the speaker of (2), and if using an alternative implies believing that it
is true, the hearer will conclude that the speaker believes that (3) is false, and that
John ate some but not all of the cake.
The general 2-step architecture we just went through follows the proposals of
Soames (1982), Hirschberg (1985/1991), Horn (1989), and more recently Sauerland
(2004), and I will make extensive use of it in discussing possible choices for the
definition of alternatives. The first step derives weak, or, using Sauerland’s terminology, primary implicatures using a conversational principle. The second step
strengthens these to secondary implicatures under certain conditions (see Fox 2006
for important modifications of this part). In terms of this architecture, the present
paper focuses entirely on the first step.
1.2 The symmetry problem
As an attempt to turn (4) into a working definition, we can first try the following
idea, based on an over-simplification of Grice, and more directly on the formulation
in Gamut (1991). The linguistic objects, / and its alternatives, are full sentences.
Using them, in the sense of the first occurrence of use, means asserting them in
discourse. Better means more informative, understood in terms of semantic
entailment. And using an alternative /0 , in the sense of the second occurrence of
use, means that it is believed to be true, relevant, and supported by evidence. This
last condition means, roughly, that /0 would be assertable in the absence of any
other alternative. It will be convenient to have a shorter way to say this. I will do this
by defining a notion of weak assertability2 :
(5)
WEAK ASSERTABILITY:
A structure / will be said to be weakly assertable by a speaker S if S
believes that / is true, relevant, and supported by the evidence.
Our version of the conversational principle will now look like this:
(6)
Conversational principle (naı̈ve version): do not assert / if there is
another sentence /0 such that both
a. ½½/0 ½½/, and
b. /0 is weakly assertable
2
The definition of weak assertability makes reference to a speaker S. Most of the time, however, no
confusion is likely to arise, and I will usually suppress this parameter.
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We now have what looks like a concrete version of the conversational principle in
(4), which we could use to reason about examples like (2) above. As noted by Kroch
(1972), however, (6) does not work. The problem, dubbed the symmetry problem
in class notes by Kai von Fintel and Irene Heim (see also Horn 2000; von Fintel and
Fox 2002), has to do with the fact that (6) allows us to reason about too many
alternatives: for any /0 that is stronger than /, and that we would like to reason
about, there is another alternative, /00 ¼ / ^ :/0 , which is also stronger than /, and
which would license an inference in the opposite direction. Combined, /0 and /00
license only ignorance inferences, contrary to fact.
To see how the symmetry problem arises in a simple example, let us return to (2)
above, repeated here:
(7)
John ate some of the cake
Assuming that the speaker obeys (6), the hearer can conclude that there was no
sentence that is strictly more informative than (7) that the speaker could have
(weakly) asserted instead. In particular, the speaker could not have asserted this
(repeated from (3)):
(8)
John ate all of the cake
(8) is more informative than (7). Assuming that (8) is relevant, the hearer may
conclude that either (8) is not supported by the evidence that the speaker had or that
the speaker believes it to be false. If (8) were the only sentence to consider in the
context of (7), and if it is plausible that the speaker has access to the relevant
evidence, the hearer could conclude that the speaker believes (8) to be false.
However, (6) does not restrict us to (8) alone. Consider (9), for example:
(9)
John ate some but not all of the cake
Like (8), (9) is also strictly more informative than (7), and the same reasoning as
above would lead the hearer to conclude that either (9) is not supported by the
evidence that the speaker has, or that the speaker believes it to be false. Again, if (9)
were the only sentence to consider, and if it is plausible that the speaker is wellinformed, the hearer could conclude that the speaker believes that (9) is false, and
that John ate all of the cake. And here is the problem: given that (7) is true, the two
alternatives (8) and (9) cannot be simultaneously false. Either John ate all of the
cake or he didn’t.3 The speaker cannot believe that (8) and (9) are both false (unless
the speaker is entertaining contradictory beliefs), and the hearer can reason that the
speaker is not opinionated with respect to at least one of the two alternatives. But for
a speaker who believes that (7) is true, being opinionated about one of the two
alternatives means being opinionated about the other, and the hearer can conclude
that the speaker is not opinionated about either. In other words, the speaker did not
3
Assuming that all presuppositions are satisfied.
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have access to the relevant evidence after all, and so was not in a position to believe
either (8) or (9). The speaker is simply ignorant of the relevant facts.
Ignorance inferences of this kind often conflict with other beliefs of the hearer.
For example, the hearer might be quite confident that the speaker was watching John
throughout the cake-eating event. And yet there is nothing deviant in (7) even in this
context. The assertion implies neither that the speaker has lost her recollection of
whether or not John ate all of the cake nor that the speaker is non-cooperative. In
fact, the standard inference in this situation is that the speaker believes that John did
not eat all of the cake. That is, actual hearers reason as if only (8) were an alternative to (7) for purposes of conversational reasoning. Fixing (6), then, will require
excluding (9) from the reasoning process.
1.3 Restricting the alternatives
1.3.1 The neo-Gricean approach
(6) said that when / is uttered, every alternative /0 should be considered. The
symmetry problem arose from the fact that for each such /0 , a symmetric /00 was
available. A widely accepted solution involves being more careful about the
alternatives that are being referred to. Each sentence / can be thought of as being
associated with a set of alternatives, Að/Þ, and it is only those alternatives that are
considered for purposes of deriving implicatures. If Að/Þ is chosen appropriately, it
will consist of those alternatives that correspond to the actual inferences that are
made. Symmetric alternatives of the kind that gives rise to unattested ignorance
inferences will be excluded. Here is a minimally revised version of (6) that can refer
to alternatives.
(10)
Conversational principle (alternative-sensitive): do not assert / if there is
another sentence /0 2 Að/Þ such that both
a. ½½/0 ½½/, and
b. /0 is weakly assertable
For (7) above, for example, the goal would now be to find a set of alternatives that
will include (8) but not (9). Of course, one would like to have a principled way to
arrive at Að/Þ for any given /. If we had to associate each sentence with an arbitrary
set of alternatives, the theory would hardly be predictive.
1.3.2 Scales
Scales, introduced by Horn (1972) and developed further by Gazdar (1979), Atlas
and Levinson (1981), and others, offer a partial solution to the problem of deriving
Að/Þ. The intuition is this: to determine that (8) is a good alternative for (7) while
(9) is not, all we have to know is that all is an alternative expression to some while
some but not all is not. Determining whether sentences containing these expressions are alternatives to each other is done mechanically, by substitution. The idea,
then, is to break down the task of deriving A, the relation between full sentences and
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their alternatives, into two components. The first component is a new relation, S
(for scale-mate-of), defined over a proper subset of linguistic objects, sometimes
referred to as scalar items (the set of scalar items is typically finite, and is often
taken to be a subset of the lexicon). The second component uses S to derive A by
substitution: if ha; a0 i 2 S, and if / is a sentence that has an occurrence of a, then
h/; /½a0 =ai 2 A, where /½a0 =a is obtained from / by replacing that occurrence of
a with a0 .4 In our example, it would suffice to specify that hsome, alli 2 S while
hsome, some but not alli 62 S.5
The scalar approach offers a way to avoid stating separately for every sentence
what its alternatives are. However, if we don’t know how scales are derived, their
usefulness will be limited. For any two expressions we would need to know in
advance whether they are scale-mates or not; otherwise, no predictions can be made.
Perhaps we can do no better than that. Gazdar (1979, p. 58), for example, concluded
that scales are ‘‘. . . in some sense, ‘given to us’’’. I will try to show that we can do
better than that, and that structural complexity provides an adequate characterization
of alternatives. This direction, though, will eventually lead us to abandon the notion
of scales.
1.3.3 Complexity and its problems
The role of complexity in communication was explored by Zipf (1949), Grice
(1989), McCawley (1978), Atlas and Levinson (1981), and Horn (1984), among
others (for more recent work, see Blutner 2000; Parikh 2000; van Rooij 2004). For
our example (7) we can state the reasoning as follows. (7) and (8) are of roughly the
same complexity, in some sense, and so they can be compared for purposes of
computing implicatures. (9), on the other hand, is more complex than (7), and is
consequently ignored. I stated the informal reasoning process in terms of comparing
whole utterances. It is easy to do the same in terms of scales: some and all are of the
same complexity, and can therefore be on the same scale, while some but not all is
more complex than some and cannot be on the same scale with it.6
Within the Gricean framework, complexity has often been related to the Maxim of
Manner. The status of this maxim, however, has remained unclear. For example, one
4
As Sauerland demonstrates, the replacement of elements by their scale-mates should not be restricted to
one-step substitutions. Rather, any sequence of replacements of elements with their scale-mates that can
take us from / to /0 will license using /0 as an alternative.
5
A further point that Sauerland makes is that scale-mate-of must be a symmetric relation: what matters
is the entailment relation between the whole sentences / and /½a0 =a and not between the scalar items a
and a0 within them. This is important for dealing with the phenomenon of scale reversal in downwardentailing contexts. The order in which the items are listed on the scales here, then, should not be taken as
significant.
6
In the literature, one often finds attempts to characterize alternatives in terms of degree of lexicalization, as distinct from brevity. Thus, Atlas and Levinson (1981, p. 44) write that ‘‘. . . to constitute a
genuine scale for the production of scalar implicatures, each item must be lexicalized to the same
degree.’’ As discussed by Matsumoto (1995), it is not always clear how one should distinguish between
brevity and lexicalization (Matsumoto himself argues that neither of these is relevant for implicature). See
Horn (2000) for further discussion on this matter, as well as for reasons to reject Atlas and Levinson
(1981)’s symmetric condition in favor of a formulation that allows using alternatives that are strictly more
lexicalized than what was uttered, much in the spirit of the current proposal.
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wonders whether a cooperative speaker should avoid providing relevant information,
as in the excluded some but not all, just in order to maintain simplicity. It is also
somewhat surprising that the supposed effects of Manner are manifest almost
exclusively in licensing alternatives of the same complexity, as with (7) and (8).
Shouldn’t we also expect to find a complex sentence having implicatures that are
based on negating simpler alternatives? As mentioned in the introduction, however,
candidates for such implicatures appear to be few and far between.
More serious concerns about the Maxim of Manner have to do with the precise
notion of complexity that is used. Certain definitions, such as syllable count and
phonetic effort, seem to yield incorrect results (see Poser 1992 for discussion).
Worse, as argued by Matsumoto (1995), some implicatures require alternatives that
are strictly more complex than the actual utterance under any reasonable definition
of complexity. If this is correct, complexity cannot be the way to go.
The argument is based on the results of asserting (11a):
(11)
a. It was warm yesterday, and it is a little bit more than warm today
(Matsumoto 1995, ex. 39, p. 44)
b. It was a little bit more than warm yesterday, and it is a little bit more
than warm today
As Matsumoto notes, uttering (11a) gives rise to the inference that it was only warm
yesterday, and not a little bit more than warm. Intuitively, this inference seems to
rely on considering the stronger alternative (11b) and concluding that it was false.
The problem, of course, is that (11b) is more complex than (11a).
1.3.4 Monotonicity
Complexity, then, appears to be on the wrong track. A more promising approach,
Matsumoto suggests, is the one that was outlined in Horn (1989), and in which the
semantic notion of monotonicity (or scalarity) plays a role.
(12)
‘‘Positive and negative quantifiers, modals, and related operators must be
represented on distinct, though related scales. There can be no single
scale on which operators like some and not all, or possible and unlikely,
can be plotted. Rather, there is one scale defined by the positive operators
and one by their negative counterparts.’’ (Horn 1989, p. 235)
(13)
‘‘Scalarity Condition: Expressions that form a Horn scale must be either
1) all positively scalar (e.g., hall, somei) or 2) all negatively scalar
(e.g., hno, fewi).’’ (Matsumoto 1995, p. 46)
Condition (13) licenses the alternatives that we have seen so far.7 It also prevents
expressions such as some but not all from being on the same scale as some, or
indeed on any scale.
7
Though see Sevi (2005) for complications with applying the notion correctly.
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In fact, one could try to further strengthen the argument for monotonicity and
against complexity, as was pointed out to me by Danny Fox (p.c.). Imagine a
complexity-based theory that would somehow allow (11b) to be used as an alternative for (11a). Such a theory, it seems, will also allow (14b) to be used as an
alternative to (14a):
(14)
a. John talked to some of the girls yesterday, and he talked to some but
not all of the girls today
b. John talked to some but not all of the girls yesterday, and he talked to
some but not all of the girls today
However, uttering (14a) does not give rise to the inference that (14b) is false. In fact,
(14a) does not seem to give rise to any inference at all. It is just an odd sentence. For
a monotonicity-based account, this is not a problem. The non-scalar some but not
all is ruled out as an alternative on semantic grounds. It seems, then, that a complexity-based approach would have to include both a special mechanism that would
license a little bit more than warm as an alternative for warm in sentences such as
(11a), and a monotonicity condition to rule out some but not all as an alternative to
some in the otherwise analogous (14a). This looks like a fairly direct argument
against the complexity approach.
2 A puzzle
Having convinced ourselves that monotonicity works and that complexity does
not, let us now look at the following sentences. In each example, uttering the
(a) sentence seems to imply that the (b) sentence is unassertable.
(15)
a. I doubt that exactly three semanticists will sit in the audience
b. I doubt that three semanticists will sit in the audience
(16)
a. If we meet John but not Mary it will be strange
b. If we meet John it will be strange
(17)
(Danny Fox, p.c.)
a. Everyone who loves John but not Mary is an idiot
b. Everyone who loves John is an idiot
By uttering (15a), for example, the speaker asserts that they find it unlikely that the
number of semanticists in the audience will be exactly three, but they also suggest
that they do not find it unlikely that the number would be at least three. Similarly,
(16a) asserts that all situations in which we meet John but not Mary will be strange,
while implying that, for all the speaker knows, there can be situations in which we
meet John and in which there is no strangeness (in those situations we would
necessarily also meet Mary, though it is conceivable that there would also be some
strange situations in which we meet both John and Mary). And (17a) suggests that,
as far as the speaker is concerned, it is possible to love John without being an idiot
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(again, as long as one also loves Mary). In each example, the (b) sentence is stronger
than the (a) one, so if we could use the (b) sentences as alternatives, we would be
able to derive the implication that the (b) sentence is not assertable as an implicature
of the (a) sentence.8 The problem is that both exactly three semanticists and John
but not Mary are non-scalar, and so the monotonicity approach predicts that they
would not be used. On the other hand, the complexity approach makes the correct
predictions here.
I will try to argue that the situation in (15)–(17) is not an accident, and that
complexity, under the right formulation, provides a correct characterization of the
alternatives, while monotonicity is wrong. To be able to make this claim I will have
to provide a concrete definition of complexity-based alternatives, and I will also
have to explain why it is that in some cases complexity appears to be wrong and
monotonicity appears to be right.
3 Proposal
3.1 Structural complexity
Certain operations simplify a structure. Viewing syntactic structures as trees (or
directed graphs, more generally, with edges representing motherhood), simplifying
operations may involve deletion (removing edges and nodes) and contraction
(removing an edge and identifying its end nodes). Other operations do not simplify but
also do not add complexity. Substitution of one terminal element for another terminal
element of the same category is an example, and more generally, substitutions of
structures for other structures given an appropriately defined substitution source. For
the moment we can think of the substitution source as the lexicon of the language,
though we will revise this once we get back to Matsumoto’s sentences in Sect. 5.9
(18)
SUBSTITUTION SOURCE (first version, to be revised in 41):
Let / be a parse tree. The substitution source for /, written as Lð/Þ,
is the lexicon of the language.
We can now say that a structure w is no more complex than / if w can be
obtained from / by a finite number of operations of the kind discussed above. Here
are the definitions.
8
That these are indeed implicatures can be seen by their cancelability. For example, the suggestion that it
is possible to love John without being an idiot disappears if we continue (17a) with ‘‘In fact, everyone
who loves John [period] is an idiot.’’ I thank an anonymous reviewer for pointing this out to me.
9
The existence of a lexicon is a contentious issue (see Marantz 1997 for discussion), and in nonlexicalist approaches such as Distributed Morphology (DM, Halle and Marantz 1993) there is no place
where words are listed to the exclusion of non-words. The use of the lexicon under the current proposal is
done only for obtaining a set of easily accessible constituents for substitution. These do not have to be
words, and the revision that will be needed to deal with Matsumoto’s examples will bring in things that
are strictly non-words. It might therefore be better to use something like the DM notion of vocabulary in
the definition of Lð/Þ, though for presentation purposes I will stick to the term lexicon.
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(19)
679
STRUCTURAL COMPLEXITY:
Let /; w be parse trees. If we can transform / into w by a finite series of
deletions, contractions, and replacements of constituents in / with
constituents of the same category taken from Lð/Þ, we will write
w . /. If w . / and / . w we will write / w. If w . / but not / . w we
will write w < /.
Using the structural relation . we can now provide a complexity-based
definition of the set of alternatives.
(20)
STRUCTURAL ALTERNATIVES:
Let / be a parse tree. The set of structural alternatives for /, written as
Astr ð/Þ, is defined as Astr ð/Þ :¼ f/0 : /0 . /g.
3.2 Using the alternatives
Having a set of alternatives is one thing. Using them is another. In the informal
discussion above I have assumed a global comparison of alternatives. This is the
traditional (neo-)Gricean approach (see Horn 1972; Gazdar 1979; Atlas and
Levinson 1981; as well as Sauerland 2004; Russell 2006; Spector 2007; Geurts 2007
for a recent defense of the global approach). Over the past few years it has been
recognized that the global approach faces non-trivial difficulties, and a localist
approach to implicatures has been developed by Landman (2000), Chierchia (2004),
and Fox (2006). It is not the goal of this paper to decide between these two
approaches. Both make crucial use of alternatives, and both are compatible with the
definition of structural alternatives in (20). I will keep discussing implicatures using
the globalist perspective of Sauerland’s 2-step architecture, outlined above, but this
is done for ease of exposition only.10
Obtaining the primary implicatures is done through the neo-Gricean principle
(10) above, where Að/Þ is now specified to be Astr ð/Þ, the structural alternatives of
/ defined in (20).
(21)
Conversational principle (with structural alternatives): do not use /
if there is another sentence /0 2 Astr ð/Þ such that both
a. ½½/0 ½½/, and
b. /0 is weakly assertable
10
Here is a brief sketch of how the current discussion could be recast in a localist system, along the lines
of Fox (2006). Implicatures, in Fox’s system, are derived using an exhaustive operator, Exh, roughly a
non-presuppositional counterpart of only. When Exh attaches to a clause /, the result is an assertion that
p ¼ s/t, the proposition denoted by /, is true, and that all the alternatives to p in a set of alternatives A
that can be safely negated (Fox’s innocently excludable alternatives, here written as IEðA; pÞ) are false:
½½½Exh/ ¼ kws ½½/ðwÞ & 8q 2 IEðA; pÞ :qðwÞ
On an account that derives implicatures using such an operator, the current proposal would translate into
restricting the set A to be a subset of the set of denotations of the structural alternatives to /. That is,
A ¼ fs/0 t j /0 2 Astr ð/Þg.
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3.3 Aside: Astr and the Maxim of Manner
Notice that (21) expresses an indirect preference for simple sentences: if /0 is
strictly simpler than / then /0 will be taken into account in computing the implicatures for /, but / will not be relevant for the implicatures of /0 . On the other
hand, there is no direct pressure to minimize structure: the fact that /0 is strictly
simpler than / does not on its own mean that /0 is preferable to / if the two are
truth-conditionally equivalent. In other words, we are using complexity as a neoGricean filter on structures and not as a preference that might correspond to the
Gricean Maxim of Manner. Let us look at what a Gricean formulation would look
like. Here again is the conversational principle we started with:
(22)
Conversational principle (informal version, repeated from (4)): do not use
a linguistic object / if there is another object /0 such that both
a. /0 is better than /, and
b. /0 could have been used
The choice of interpreting better in terms of semantic entailment in (22) was a
simplification, based on one part of Grice’s Maxim of Quantity. To incorporate
Manner, we could combine semantic entailment with our notion of structural
complexity . . Using - for at-least-as-good-as, we could write this as follows:
(23)
- :¼ fð/; wÞ j / . w ^ ½½/ ½½wg
We could then take better to mean the irreflexive (defined as - n -1 ), changing
our neo-Gricean (21) into a Gricean principle11 :
(24)
Conversational principle (Gricean): do not assert / if there is an alternative
/0 such that
a. /0 /
b. /0 is weakly assertable
Treating complexity as part of the comparison, as in (24), is perhaps more natural
from a globalist, pragmatic perspective than using it as part of the filtering mechanism, as in (21).12 More significantly, the two approaches make distinct predictions. If s/t ¼ s/0 t then under (21) the weak assertability of /0 will never affect the
assertability of /. Under (24), if /0 is strictly less complex than /, the weak
assertability of /0 will prevent / from being assertable. Following these predictions
will take us beyond the scope of the current paper, as does the evaluation of other
11
More generally, we could start with any set of pre-orders over the domain of structures U (that is,
reflexive, transitive relations over U U), fa ga2I . In addition to semantic entailment and structural
complexity, one can use pre-orders that correspond to obscurity, appropriateness, politeness, and so on.
The relation
of at-least-as-good-as can then be defined as the intersection of these pre-orders:
T
- :¼ a a . The rest proceeds as above.
12
See Fox (2006) and Russell (2006) for considerations regarding the role of formal alternatives in
conversational reasoning.
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Structurally-defined alternatives
681
choices that could have been made in the formulation of the conversational principle. For the examples that we will discuss below, the differences between the two
formulations will not matter. I will keep using the neo-Gricean (21), mostly in order
to facilitate the comparison with other proposals.
4 Examples
4.1 Some, all, some but not all
Let us start by verifying that our definitions can handle the simple (7) with which we
started, repeated here.
(25)
a. / ¼ John ate some of the cake
b. /0 ¼ John ate all of the cake
c. /00 ¼ John ate some but not all of the cake
Assuming that all and some are of the same syntactic category, and assuming that
both are in the lexicon, we can substitute one for the other and get from / to /0
(or from /0 to /) without adding complexity. That is, /0 /, and by definition,
/0 2 Astr ð/Þ. Consequently, when / is uttered, /0 is evaluated by (21), which tells
us that / should not be uttered if (i) s/0 t s/t, and (ii) /0 is weakly assertable.
Assuming that the speaker obeys (21), then, the utterance of / means that at least
one of (i) and (ii) does not hold. (i) is true, so (ii) must be false. That is, /0 is
believed to be not true, not relevant, or not supported by the evidence.
/00 is also stronger than /, and we saw that if it is not excluded from being an
alternative it gives rise to the symmetry problem. Under the current definitions, /00
is successfully excluded. The alternatives of / are only structures that are at most as
complex as /, and /00 is not such a structure: there is no way in which we can
transform / into /00 by the operations relevant for structural complexity listed in
(19). In fact, /00 is strictly more complex than /. (21) does not consider /00 , and so
no inference is made.
Summing up, uttering / gives rise to the inference that it is not the case that /0 is
weakly assertable, and it does not give rise to any inference with respect to /00 .
Assuming that no other alternatives are available, the only primary implicature of /
is that /0 is not weakly assertable. This now feeds the second process in which the
primary implicatures are strengthened. Under the appropriate assumptions, the
secondary implicature is that /0 is false. In total, we obtain the desired inference that
John ate some but not all of the cake.
4.2 ‘or’, ‘and’, L, R
Disjunction gives rise to two main inferences: that only one of the disjuncts is true
(exclusiveness), and that the speaker does not know which one it is (ignorance).
We start with exclusiveness. Uttering /, for example, usually implies that it is not
the case that John ate both the apple and the pear. That is, / implies that /0 is false.
123
682
(26)
R. Katzir
a. / ¼ John ate the apple or the pear
b. /0 ¼ John ate the apple and the pear
c. /00 ¼ John ate the apple or the pear but not both
To obtain the exclusive reading of / we must make sure that /0 is an alternative, but
the symmetric /00 is not. Under the current approach, this is the result of the same
structural consideration that prevented the symmetry problem elsewhere: /0 is at
most as complex as /, while /00 is strictly more complex. So far, nothing special.
Turning to the ignorance inference, it was observed by Grice (1989) (see also
Gazdar 1979, among others) that disjunction usually implies that the speaker does
not know which of the disjuncts holds. In / above, the speaker can be taken not to
be in a position to say which of the two fruits John ate. A proposal for deriving the
ignorance inference has been developed by Sauerland (2004), and I will adopt
the proposal in its essentials. I will show, however, that one particular aspect of the
proposal can be simplified: the effects of a seemingly stipulative definition of scales
are directly predicted from our complexity-sensitive alternatives.
As Sauerland notes, the ignorance inference for disjunction can be obtained if /
has w0 and w00 as alternatives.
(27)
a. w0 ¼ John ate the apple
b. w00 ¼ John ate the pear
Semantically, w0 and w00 are both strictly stronger than /. If w0 and w00 are alternatives to /, we may conclude that neither of them was weakly assertable. Since
both seem to be relevant in situations in which / is, as was /0 above, we obtain the
following primary inferences:
(28)
a. It is not the case that the speaker believes that /0
b. It is not the case that the speaker believes that w0
c. It is not the case that the speaker believes that w00
As before, these primary inferences now feed the process of strengthening, resulting
in secondary implicatures. For (28a) strengthening is straightforward, yielding the
inference that the speaker believes that /0 is false. For (28b) and (28c), on the other
hand, no strengthening takes place, since it is impossible to strengthen both inferences simultaneously without contradicting the original assertion /. See Sauerland
(2004) and Fox (2006) for details and discussion.
The key to obtaining the correct inferences for / is the alternatives. Specifically,
we need /0 ; w0 , and w00 to be alternatives, while /00 must be excluded. Structurally
defined alternatives, using . , derives precisely these alternatives. Notice, however,
that there was something new in this example. So far, the good alternatives were
always of the same complexity as the original utterances. The excluded structures
were strictly more complex. Here, for the first time, we needed to use alternatives
that were strictly less complex than the original structure. Relatedly, this is also the
first example in which our proposal behaves differently from a scale-based
approach. For a scale-based approach, including /0 and excluding /00 is fairly
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Structurally-defined alternatives
683
straightforward: or and and are standardly assumed to be scale-mates, and there is
no scale which allows us to get to /00 . Deriving w0 and w00 , on the other hand, is less
simple. The problem, discussed by Sauerland (2004), is this: for any disjunction
p _ q we would like to have both p and q as alternatives. The relation scale-mate-of,
however, is really an equivalence relation. This would mean that for any p and any q
we would have p _ q as an alternative, which in turn would mean that p and q are
scale-mates of each other, regardless of what they are. But if every sentence is an
alternative to any sentence, all we would get is ignorance inferences, and no real
(secondary) implicatures would ever arise.
Sauerland’s solution is to posit two binary connectives, L(eft) and R(ight), which
are scale-mates to _ and ^. L returns its left argument, and R returns its right
argument: s/Lwt ¼ s/t ¼ swR/t. With these new scale-mates, / _ w will always
have both /Lw and /Rw as alternatives, simulating the effect of using each disjunct
as an alternative. Importantly, this does not actually turn the disjuncts into alternatives: / is not the same as /Lw (or wR/); consequently, the problem of arbitrary
sentences becoming alternatives of each other disappears.13
The introduction of L and R and the conditions on them are somewhat stipulative.
Our structural alternatives derive the effect of these connectives using nothing more
than what was needed to obviate the symmetry problem with some, all, and some
but not all.
4.3 Strictly simpler alternatives
Under the current proposal, we should expect to find effects similar to those of L and
R but in other domains. That is, whenever we have a complex structure /, we
predict that any simplification of / will be an alternative to it. For example, a
structure that contains a modified noun phrase should have as an alternative the
same structure but without modification. Normally, however, we see no such
effects, which is one of the reasons for skepticism towards complexity-based
theories. But there is a reason why such inferences are rare. In most cases, a
sentence with a modified noun phrase will (asymmetrically) entail the variant with
no modification:
(29)
a. / ¼ A tall man came to the party
b. /0 ¼ A man came to the party
Since / is stronger than /0 , we cannot negate /0 without contradicting the assertion.
To test for complexity effects in implicatures we need to reverse the entailment
relations. Under downward-entailing operators, then, we will expect to find implicatures that are based on complexity. In fact, we already saw some such cases in
(15) above. Here are some further examples. In each case, the (a) sentence implies
that the (b) sentence is not assertable.
13
Or, at least, it disappears if we assume that uttering / can never be mis-parsed as /Lw or as wR/.
Sauerland addresses this concern by a prohibition against using L and R in speech, an effect that he
attributes to the Maxim of Manner.
123
684
R. Katzir
(30)
a. / ¼ If any tall man comes to the party, he will be disappointed
b. /0 ¼ If any man comes to the party, he will be disappointed
(31)
a. w ¼ Every candidate who sang was elected
b. w0 ¼ Every candidate was elected
(32)
a. n ¼ John doubts that many dogs with long tails will be sold
b. n0 ¼ John doubts that many dogs will be sold
As with disjunction, these inferences pose a challenge to scale-based approaches. If
we wanted to capture the implicatures above in terms of scales, we would probably
say that tall man is a scale-mate of man, candidate who sang is a scale-mate of
candidate, and dogs with long tails is a scale-mate of dogs (and perhaps more
generally that any noun phrase that contains a modifier is a scale-mate of the same
noun phrase but without the modifier). But this would predict that in upwardentailing contexts we would get implicatures in the opposite direction, from the
simple structure to the complex one. This prediction is not borne out: in none of the
examples below does the (a) sentence imply that the (b) sentence is false.14
(33)
a. / ¼ A man came to every party
b. /0 ¼ A tall man came to every party
(34)
a. w ¼ Each reporter talked to a candidate
b. w0 ¼ Each reporter talked to a candidate who sang
(35)
a. n ¼ John is sure that many dogs will be sold
b. n0 ¼ John is sure that many dogs with long tails will be sold
The (a) sentences suggest nothing about their (b) counterparts. Under the direct
approach, using structural alternatives, the absence of inferences about more
complex structures, as in (33), is predicted.
As we saw in Sect. 2 above, the same pattern holds also for non-monotonic noun
phrases. Here are the examples again.
(36)
a. I doubt that exactly three semanticists will sit in the audience
b. I doubt that three semanticists will sit in the audience
(37)
a. If we meet John but not Mary it will be strange
b. If we meet John it will be strange
(38)
(Danny Fox, p.c.)
a. Everyone who loves John but not Mary is an idiot
b. Everyone who loves John is an idiot
14
The use of universal operators in these examples guards against the possibility that inferences would
be blocked because of symmetry. See Sect. 5.2 for discussion.
123
Structurally-defined alternatives
685
In (36) the triggering elements, exactly three semanticists and John but not Mary,
were not monotonic, though their alternatives were. It is just as easy for the alternatives themselves to be non-monotonic:
(39)
a. I doubt that exactly three semanticists and exactly two syntacticians
will sit in the audience
b. I doubt that exactly three semanticists will sit in the audience
In total, then, we have a system that dispenses with the notion of scale, derives
the effect of L and R for disjunction, and makes new (and correct) predictions about
a family of inferences that arise in downward-entailing contexts, inferences that
pose a challenge for the monotonicity-based approach and for the very notion of
scale.
5 Complex alternatives: redefining the substitution source
It is now time to go back to Matsumoto’s argument against the use of complexity in
implicatures. The argument was based on example (11a) above, repeated here:
(40)
a. / ¼ It was warm yesterday, and it is a little bit more than warm today
b. /0 ¼ It was a little bit more than warm yesterday, and it is a little
bit more than warm today
Our task is to allow /0 to be an alternative for / even though it is strictly more
complex. I would like to suggest that what licenses /0 as an alternative is the fact
that, in some sense, its building blocks are already present in /. To obtain /0 from /
what we need is to substitute a little bit more than warm for warm in /. The
structure for a little bit more than warm is not in the lexicon, which prevented us
from performing the relevant substitution so far, but it is there as part of the right
conjunct of /. If we could enrich the substitution source with the subtrees of the
current utterance, we would get what we want. Here is the definition15 :
(41)
SUBSTITUTION SOURCE (final version):
Let / be a parse tree. The substitution source for /, written as Lð/Þ is the
union of the lexicon of the language with the set of all subtrees of /
With our revised definition for Lð/Þ in place we can now derive the desired
implicature. Since a little bit more than warm is a subtree of /, we can use it for
15
Not all speakers share Matsumoto’s intuitions about (40a). Among those who do, some seem to require
a special prosodic marking of warm and a little bit more than warm, similar perhaps to the marking
discussed by Geurts (2007) in the context of Levinson (2000)’s examples. I have nothing interesting to
say about the source of this inter-speaker variation. For now I will assume that speakers differ in whether
they use only the lexicon as their substitution source, as in our original definition, or whether they allow
also substitutions outside the lexicon, as in (41), possibly subject to the requirement that substitutions
outside the lexicon be prosodically marked in the relevant way.
123
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R. Katzir
substitution and obtain /0 , which means that /0 . /, appearances to the contrary
notwithstanding. Semantically, we may assume that sa little bit more than
warmt swarmt. A speaker who obeys (21) will therefore not assert / if /0 is
weakly assertable. Assuming that there are no symmetric alternatives to consider,
strengthening may now proceed, giving rise to the inference that it was warm but
not a little bit more than warm yesterday, and it is a little bit more than warm today.
5.1 Concern I: arbitrariness
We have thus derived the inferences which Matsumoto used against complexitybased approaches. Though the change to our original system was not big, it does
look like a hack, added to simulate Matsumoto’s results without giving up on
structural comparisons. We could, perhaps, justify our move by saying that the
substitution source is the collection of objects that are available for further syntactic
operations. Those include the lexicon, used as a source for the terminal elements in
the derivation, but they also include any constituent of the current structure (and
maybe nearby structures as well)16 that could be used for movement, pronominalization, the satisfaction of anaphoricity requirements on focus and ellipsis, and so
on. Perhaps. Fortunately, we do not have to rely on such considerations. As it
happens, the two proposals are not equivalent. They make different predictions, and
it is the predictions of the current one that are borne out.
Under the current proposal the inference in cases like (40a) depends on the
presence of the more complex alternative somewhere within the structure. If it is not
there, no implicature will be generated. Under a scale-based approach, on the other
hand, once a scale-mate always a scale-mate. (40a) would teach us that warm and
a little bit more than warm are scale-mates; after that we would expect to find
similar inferences elsewhere, regardless of whether a little bit more than warm is
present or not. The facts seem to support the complexity approach. When a little bit
more than warm is absent, no inference about yesterday’s temperature is made:
(42)
It was warm yesterday, and it is cold today
(43)
It was warm yesterday, and it is hot today
16
That nearby structures are also available for substitution can be seen from the possibility of replicating
Matsumoto’s results across discourse. An anonymous reviewer, for example, offers the following
exchange, to be imagined as part of a long-distance phone call:
A: I love late spring. It’s warm here.
B: It’s a little bit more than warm here. I’m inside with the air conditioning.
As in Matsumoto’s example, B’s utterance implies that it is warm, but not a little bit more than warm on
A’s side. A similar point can be made using the following exchange, from Gazdar (1979), where B’s
answer is taken to imply that B’s mother is not well:
A: Is your mother well and back?
B: Well, she’s back, yes.
A: She’s not well then.
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Structurally-defined alternatives
687
It is also worth noting that the current approach, differently from the scale-based
one, predicts that more or less any other inference can be triggered if the relevant
material is already part of the structure.17 This, too, seems to be correct: uttering
(44) implies that yesterday it was just warm, not warm and sunny with gusts of
wind.
(44)
It was warm yesterday and it is warm and sunny with gusts of wind today
5.2 Concern II: non-scalar items
Matsumoto’s example was meant as an argument against using complexity in the
computation of implicatures. Upon closer inspection, as we just saw, this example
turned out to teach us about the exact notion of complexity that is involved. It also
constitutes another argument against the notion of scale. But should the definition of
alternatives ignore any non-structural considerations? The patterns that we discussed suggest two conflicting conclusions. On the one hand, we observed that nonmonotonic elements can trigger implicatures (as in (15) above), and that the
alternatives themselves can be non-monotonic (as in (39)). Those implicatures are
often difficult to find, since the simpler alternatives are usually weaker, but in
downward-entailing contexts the implicatures surfaced. On the other hand, we have
encountered other data that seem to suggest the opposite: when Matsumoto’s
example is modified to involve non-monotonic elements instead of the original
monotonic ones, no inference arises. The relevant example was (14a) above,
repeated here.
(45)
a. John talked to some of the girls yesterday, and he talked to some but
not all of the girls today
b. John talked to some but not all of the girls yesterday, and he talked to
some but not all of the girls today
Uttering the distinctly odd (45a) does not give rise to any inference about (45b), as
we saw. Should we then find a way to incorporate monotonicity as a condition in
cases like (45a)? I think not. The problem is one of symmetry. Notice that (45b) is
not the only alternative to (45a). The following is also a stronger alternative:
(46)
John talked to all of the girls yesterday, and he talked to some but not all
of the girls today
Combining (45b) and (46), all we get for (45a) is an ignorance inference.18 In this
case, then, monotonicity and brevity make similar predictions. It is possible to tease
these predictions apart, though. If monotonicity holds, it should hold in all cases. If
the problem is one of symmetry, embedding the structure in a context where the two
17
Subject to considerations of symmetry, as we discuss immediately below.
18
Though I do not know why the result is odd.
123
688
R. Katzir
symmetric alternatives do not exhaust the space of possibilities will solve the
problem, and an implicature is predicted to arise. What we should do, then, is
modify (45a) using a universal operator, as in (47). The two symmetric alternatives,
no longer exhausting the possibilities, are listed in (48).
(47)
Yesterday, every boy talked to some of the girls, and today every boy
talked to some but not all of the girls
(48)
a. Yesterday, every boy talked to all of the girls, and today every boy
talked to some but not all of the girls
b. Yesterday, every boy talked to some but not all of the girls, and today
every boy talked to some but not all of the girls
Judgments about (47) are far from clear. To the extent that the sentence is at all
acceptable, though, it seems to imply that yesterday at least one boy talked to all of
the girls and that at least one boy didn’t. Similar effects arise with other non-scalar
items. For example, (49) suggests that yesterday John was required to talk to at least
three girls, and that he could have satisfied this requirement by talking to exactly
three girls or by talking to more.19
(49)
Yesterday, John was required to talk to three girls, and today John was
required to talk to exactly three girls
(50)
a. Yesterday, John was required to talk to four girls, and today John was
required to talk to exactly three girls
b. Yesterday, John was required to talk to exactly three girls, and today
John was required to talk to exactly three girls
It appears, then, that monotonicity plays no role at all in implicatures. It restricts
neither the triggers nor the alternatives, as we discussed in the context of strictly
simpler alternatives in Sect. 4.3, and now we just saw further evidence that it does
not restrict the alternatives. The only relevant factor, it seems, is structure.
6 Summary
We started with the symmetry problem, which arises when every stronger alternative can be negated, as in the na€ıve conversational principle (6), repeated here.
(51)
Conversational principle (na€ıve version, repeated from (6)): do not assert /
if there is another sentence /0 such that both
a. /0 is more informative than /, and
b. /0 is weakly assertable
19
I use a modal rather than a quantifier over individuals as the universal operator in (49) following a
suggestion by Danny Fox, p.c., who points out that modals help sharpen the judgments about the
inferences in these sentences. I have no account for the difference between the operators.
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Structurally-defined alternatives
689
We followed the neo-Gricean idea of restricting the set of alternatives, though we
saw evidence against the more familiar implementations in terms of scales and
monotonicity. Instead, a structural condition on alternatives seemed to work: the
alternatives of / are all the structures that are at most as complex as /. We noticed
that in many cases, simpler alternatives are also weaker, and hence irrelevant for the
computation of implicatures. When the simpler alternatives are stronger, as with
disjunction or with weak alternatives in downward-entailing contexts, the predicted
inferences arise. Moreover, we found inferences of this kind that arise with respect
to non-monotonic expressions. Our definitions made no reference to the notion of
scales, and in fact we noticed that deriving the alternatives directly helped us avoid
some of the challenges that are faced by accounts that use that notion. Finally, in
addressing Matsumoto’s argument against complexity, we had to revise our definition of substitution source. This revision led to certain predictions that were
different from those made by Matsumoto’s account. As it turned out, the correct
predictions were the ones of the current account, providing an additional argument
in favor of the complexity-based approach and against the use of scales and
monotonicity.
Acknowledgements I thank Asaf Bachrach, Emmanuel Chemla, Gennaro Chierchia, Paul Egré, Kai von
Fintel, Irene Heim, Sabine Iatridou, Ivona Kučerová, Benjamin Spector, and Raj Singh, as well as two
anonymous L&P reviewers and Polly Jacobson, for helpful comments and suggestions. Special thanks go
to Danny Fox. Earlier versions were presented at SNEWS, September 30, 2006, at Tel-Aviv, January 4,
2007, and at the MIT-France Workshop on Presuppositions and Implicatures, Paris DEC-ENS, May 29,
2007, and I thank the audiences there for their comments.
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